To analyze active particles in confinement in the underdamped regime, we
consider the Kramers equations [1]} for the evolution of \(\rho ({\bf r},{\bf v},{\bf f},t)\) :
\(\frac{\partial \rho }{\partial t} &=& -{\bf v} \cdot \mbox{$\nabla $}_{r}\rho - \frac{1}{\tau _r} \mbox{$\nabla $}_v \cdot \left[ \left( \mu {\bf F} - {\bf v} + \mu {\bf f} \right) \rho \right] + \frac{D}{\tau _r^2} \nabla _v^2 \rho \nonumber \\&+&\frac{1}{\tau }{\left\lbrace \begin{array}{ll}\frac{1}{2\pi } \int _0^{2\pi } d\theta _f\, \rho - \rho , & \text{RTP} \\\frac{\partial ^2\rho }{\partial \theta _f^2}, & \text{ABP} \\\mbox{$\nabla $}_{\!\! f} \cdot {\bf f} \rho + \frac{D_f}{\tau \mu ^2} \mbox{$\nabla $}_f^2 \rho , & \text{AOUP}.\end{array}\right.}\) 
